The present invention relates to compensation for dispersion of fluoride based fiber lasers, and more particularly to a method for compensation for dispersion of fluoride based fiber lasers in the second optical window.
In fiber lasers, fluoride glass materials (ZBLAN) are used in the second optical window in place of quartz-based glass. In general, Pr3+ is used as the dopant. See:
[1] Ohishi, Y.; Kanamori, T.; Takahashi, S.: xe2x80x9cPr3+-doped fluoride single-mode fiber laser.xe2x80x9d IEEE Photonics Technol. Lett., 3 (1991), 688;
[2] Hodel, W.: xe2x80x9cAufbau eines Festkxc3x6rperlasers im 1300nm Wellenlxc3xa4ngenbereichxe2x80x9d [Structure of a Solid Body Laser in the 1300 nm Wavelength Range]. Zwischenbericht zum Forschungsauftrag Nr. 283 der GD PTT Uni Bern, Nov. 30, 1993; and
[3] Dxc3x6ring, H.; Peupelmann, J.; Wenzel, F.: xe2x80x9cPr3+-doped 1.3 xcexcm fiber laser using direct coated dichroitic mirrors.xe2x80x9d Electr. Lett. 31 (1955) 13, 1068.
These fibers have a zero dispersion wavelength in the third optical window; in the second optical window, the dispersion is strongly negative. If the fiber laser is used as a soliton source or as a radiation source for ultrahigh-bit-rate linear transmission systems, the time width of the pulses must be as small as possible ( less than 5-10 ps). With increasing dispersion, the pulse broadens increasingly. It is desirable to compensate for the dispersion of the fluoride glass fiber. The simplest and most effective option is to link the active fiber to a dispersion compensating (DC) fiber; the dispersion values D(xcex) in ps/km nm should be on the same order of magnitude, but naturally must have opposite signs. Both fibers are components of the fiber resonator. From this it follows that the DC fiber should be as short as possible to minimize resonator losses.
The time half-value width with respect to time (tp) of fiber lasers can be calculated using the Kuizenga-Siegman theory and its expansion by Geister. See;
[4] Kuizenga, D. J., Siegman, A. E.: xe2x80x9cFM and AM Mode-Locking of the Homogeneous Laser - Part I: Theory.xe2x80x9d IEEE J. Quant. Electr. 6 (1970), 694 ; and
[5] Geister, G. xe2x80x9cIntegrierte optische Modulation von Neodym-Faserlasernxe2x80x9d [Integrated Optical Modulation of Nyodymium Fiber Lasers]. Fortschritts-Berichte VDI Reihe 10 (1990) 140, 1, 102.                               τ          p                =                                                                                                                        2                      ⁢                                              2                                            ⁢                      ln                      ⁢                                              xe2x80x83                                            ⁢                      2                                                        π                                ⁡                                  [                                      1                                                                  f                        m                        2                                            ⁢                                              δ                        c                                                                              ]                                                            1                /                4                                      ⁡                          [                                                                    (                                                                                                                        λ                            2                                                    ⁢                                                      L                            a                                                                                                    2                          ⁢                                                      xe2x80x83                                                    ⁢                          π                          ⁢                                                      xe2x80x83                                                    ⁢                          c                                                                    ⁢                                              xe2x80x83                                            ⁢                      D                                        )                                    2                                +                                                      (                                          g                                                                        π                          2                                                ⁢                        Δ                        ⁢                                                  xe2x80x83                                                ⁢                                                  f                          2                                                                                      )                                    2                                            ]                                            1            /            8                                              (        1        )            
The half-value width tp is thus dependent on the modulation frequency fm, the modulation index xcex4c, the laser wavelength xcex, length La of the active fiber, gain coefficient g, spectral half-value width xcex94xcex of the fluorescence spectrum (corresponding to xcex94f) and the chromatic dispersion D of the fiber. The modulation frequency, modulation index, laser wavelength, gain coefficient, and fluorescence spectrum are either constants of the material or are specified parameters and thus cannot be manipulated. Fiber length La is optimized for laser operation; thus it is also a constant. Accordingly, the only parameter which can be manipulated is chromatic dispersion D, which in the most favorable case is brought to zero to achieve minimal pulse widths. This means that the dispersion of the active fiber must be compensated for. This can be achieved though the use of chirped Bragg fiber grid as the laser mirror. In principle, input mirrors or output mirrors can be implemented as Bragg grids. If grids are used for both laser mirrors, they must have the same Bragg wavelength. Reflectivity, in particular that of the output mirror, must be maintained very precisely since otherwise the laser threshold will be elevated and the output power will be reduced. According to the Kuizenga-Siegman-Geister theory (see equation (1)) the half-value width with respect to time xcfx84p increases with decreasing fluorescence bandwidth xcex94f. If the spectral width of the chirped Bragg grid is smaller than the fluorescence bandwidth, the latter will limit the minimum pulse width. To reduce the pulse width to the theoretical limit, a spectral grid width of around 60 nm would be required. The technology of grid manufacture, however, currently is by far not sufficiently advanced for the manufacture of chirped Bragg fiber grids with defined reflexivity and adequately large spectral bandwidth.
In contrast to this, the present invention compensates for the dispersion through the addition of a dispersion compensating fiber in the resonator. In high-bit-rate systems, dispersion compensation through DC fibers is a conventional method in the third optical window. See [6] Vengsarkar, A. M.; Miller, A. E.; Gnauck, A. H.; Reed, W. A.; Walker, K.: OFC 94, Technical Digest ThK2, San Jose, 1994, 225.
For this, compensation fibers with high negative dispersion are needed in the third optical window in order to compensate for the high positive dispersion of the standard fibers. In [7] Boness, R.; Nowak, W.; Vobian, J.; Unger, S.; Kirchhof, J: xe2x80x9cTailoring of dispersion-compensation fibres with high compensation ratios up to 30. xe2x80x9d Pure Appl. Opt. 5 (1995), 333, an iterative, inverse method for determining parameters and optimizing profiles is introduced which is provided for DC fibers with high negative dispersion in the third optical window.
To determine the corresponding parameters of these special fibers, an inverse problem must be solved. For some predefined dispersion values, this is possible only by using a multivariable optimization method. See [8] Ortega, J. M.; Rheinboldt, C.: xe2x80x9cIterative Solution of Nonlinear Equations in Several Variablesxe2x80x9d (New York, Academic 1970).
It is necessary to minimize the generalized standard deviation, which is expressed as follows:                                                                         F                ⁡                                  (                  x                  )                                            =                              xe2x80x83                            ⁢                                                                    ∑                                          i                      =                      1                                        n                                    ⁢                                                            b                      ⁡                                              (                                                                              D                            ⁡                                                          (                                                                                                λ                                  i                                                                ⁢                                x                                                            )                                                                                -                                                      D                            i                                                                          )                                                              2                                                  +                                                                            b                                              n                        +                        i                                                              ⁡                                          (                                                                                                    λ                            c                                                    ⁡                                                      (                            x                            )                                                                          -                                                  λ                          c0                                                                    )                                                        2                                +                                                                                                        xe2x80x83                            ⁢                                                b                                      n                    +                    2                                                  (                                                                            W                      f                                        ⁡                                          (                                              x                        ,                                                                              λ                            0                                                    -                                                      W                            f0                                                                                              )                                                        2                                                                                        (        2        )            
with respect to the unknown vector x. x=vector of the relative index of refraction 2xcex94i (1 xe2x89xa6Ixe2x89xa6m) and of the fiber radius, thus dim(x)=m+1, generally with mxe2x89xa64.       2    ⁢          xe2x80x83        ⁢          Δ      i        :=                    n        i        2            -              n        c        2                    n      c      2      
with nc=refraction index of the outer cladding, in general 1.4573. Di: specified discrete dispersion values at xcexi (1xe2x89xa6Ixe2x89xa6n), with nxe2x89xa7dim(x). xcexc0=specified cutoff of LP11 mode, w[symbol]0 of the specified Petermann-II-radius, bi=suitable positive weighting factors.
To minimize the functional equation (2), the Levenberg-Marquardt algorithm [8] (damped Gauss-Newton method) is used. This program is based on a very stable and exact method for computing dispersion on the basis of the scalar wave equation, in which numeric differentiations are avoided [7].
In the case of the fluoride glass based fiber laser in the second optical window, the high negative dispersion at xcex=1300 nm must be compensated for with correspondingly high positive dispersion. Such fibers were not known in the past; the same also applies to fiber profiles which accomplish the desired result.
A method according to the present invention is based on manipulating a compensation fiber so that through it, the high negative dispersion of a fiber laser can be compensated for in the second optical window.